Prove inequality for x^n - y^n with n natural

[MathematicalPhysicist]

i need to prove that if x>y>0 and n>1 n natural then:
n(x-y)y^(n-1)<x^n-y^n<n(x-y)x^(n-1)

i tried almost everything from n(x^n-y^n)/x-y>(x^n-y^n)/x-y>1

but to get nothing, can someone help?

 

[HallsofIvy]

You know, do you not, that xⁿ- yⁿ= (x- y)(x^n-1+ x^n-2y+ ...+ xy^n-2+ y^n-1)?

 

[MathematicalPhysicist]

well thanks.